Theorem fv2 3715

Description: Alternate definition of function value. Definition 10.11 of [Quine] p. 68.

Hypothesis

Ref	Expression
fv2.1	|- A e. V

Assertion

Ref	Expression
fv2	|- (F` A) = U.{x | A.y(AFy <-> y = x)}

Distinct variable groups:   x,y,A   x,F,y

Proof of Theorem fv2

Step	Hyp	Ref	Expression
1		df-fv 3194	. 2 |- (F` A) = U.{x | (F"{A}) = {x}}
2		dfcleq 1469	. . . . 5 |- ((F"{A}) = {x} <-> A.y(y e. (F"{A}) <-> y e. {x}))
3		dfima2 3401	. . . . . . . . . 10 |- (F"{A}) = {y | E.x e. {A}xFy}
4	3	abeq2i 1568	. . . . . . . . 9 |- (y e. (F"{A}) <-> E.x e. {A}xFy)
5		df-rex 1648	. . . . . . . . 9 |- (E.x e. {A}xFy <-> E.x(x e. {A} /\ xFy))
6	4, 5	bitr 173	. . . . . . . 8 |- (y e. (F"{A}) <-> E.x(x e. {A} /\ xFy))
7		elsn 2418	. . . . . . . . . 10 |- (x e. {A} <-> x = A)
8	7	anbi1i 481	. . . . . . . . 9 |- ((x e. {A} /\ xFy) <-> (x = A /\ xFy))
9	8	exbii 1050	. . . . . . . 8 |- (E.x(x e. {A} /\ xFy) <-> E.x(x = A /\ xFy))
10		fv2.1	. . . . . . . . 9 |- A e. V
11		breq1 2618	. . . . . . . . 9 |- (x = A -> (xFy <-> AFy))
12	10, 11	ceqsexv 1832	. . . . . . . 8 |- (E.x(x = A /\ xFy) <-> AFy)
13	6, 9, 12	3bitr 177	. . . . . . 7 |- (y e. (F"{A}) <-> AFy)
14		elsn 2418	. . . . . . 7 |- (y e. {x} <-> y = x)
15	13, 14	bibi12i 609	. . . . . 6 |- ((y e. (F"{A}) <-> y e. {x}) <-> (AFy <-> y = x))
16	15	albii 998	. . . . 5 |- (A.y(y e. (F"{A}) <-> y e. {x}) <-> A.y(AFy <-> y = x))
17	2, 16	bitr 173	. . . 4 |- ((F"{A}) = {x} <-> A.y(AFy <-> y = x))
18	17	abbii 1573	. . 3 |- {x | (F"{A}) = {x}} = {x | A.y(AFy <-> y = x)}
19	18	unieqi 2507	. 2 |- U.{x | (F"{A}) = {x}} = U.{x | A.y(AFy <-> y = x)}
20	1, 19	eqtr 1493	1 |- (F` A) = U.{x | A.y(AFy <-> y = x)}

Syntax hints:   <-> wb 146   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  E.wex 979  {cab 1462  E.wrex 1644  Vcvv 1808  {csn 2406  U.cuni 2499   class class class wbr 2615  "cima 3169  ` cfv 3178

This theorem is referenced by:  elfv 3717

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-xp 3180  df-cnv 3182  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fv 3194

Copyright terms: Public domain